The invention relates to a method for producing a stationary (intense) wave field of arbitrary longitudinal shape and high transverse localization.
Since many years it has been known that suitable frequency superpositions of Bessel beams, with the same phase velocity, yield localized wave pulses. Such pulses possessed the surprising characteristic of resisting the diffraction effects during their propagation.
In U.S. Pat. No. 5,081,995 Lu has shown that it is possible to produce a non-diffracting series of (acoustic) pulses by using a piezoelectric transducer consisting in a series of rings. The piezoelectric transducer produces ultrasonic pulses used to increase the field resolution. The use of annular transducers is common, since axial symmetry is convenient for reducing diffraction in the generated wave fields.
Optical laser-light Bessel beams have been generated by Durnin et al. with a very simple experimental apparatus comprising a laser beam source, an annular slit, and a lens. A gaussian beam generated by the laser beam is transformed into a Bessel beam which is highly non-diffracting.
It would be of high technical relevance if a wave field could be generated, possessing not only a highly non-diffracting behaviour (i.e., being radially confined relatively to the direction of propagation), but also being confined within a predetermined space-region limited both in its direction of propagation and in its transverse direction. In this situation, inside the interval 0<=z<=L, at least one 3-dimensional space-regions can be defined, having a predetermined size L1<=z<=L2 within the said interval 0<=z<=L, so that the field is therein confined, and negligible outside it. Of course, since the method uses Fourier-type transformations, the same pattern a priori gets repeated inside each successive interval L<=z<=2L, etc., unless the depth-field of each Bessel beam entering the superposition considered in the disclosed method is chosen to be not much gratee than L.
Document QING CAO ET AL: “Axially symmetric on-axis flap-top beam” JOURNAL OF THE OPTICAL SOCIETY OF AMERICA A (OPTICS, IMAGE SCIENCE AND VISION) OPT. SOC. AMERICA USA, vol. 17, no. 3, March 2000 (2000-03), pages 447-455, XP002305056 ISSN:0740-3232, describes a method for building a flat-top beam which is a square-shaped wave field having a width L. The starting point of the disclosed method is the Fourier integral of the Fourier Transform
      φ    ⁡          (      S      )        =            ∫              -        ∞            ∞        ⁢                                        I            g                    ⁡                      (                          o              ,              z                        )                          ⁢                  exp          ⁡                      (                          ⅈ2π              ⁢                                                          ⁢              Sz                        )                          ⁢                  ⅆ          z                    and its inverse function
      ∫          -      ∞              -              S        i              ⁢            φ      ⁡              (        S        )              ⁢          exp      ⁡              (                              -            ⅈ2π                    ⁢                                          ⁢          Sz                )              ⁢                  ⅆ        S            .      This method making use of the traditional Fourier Integrals the disclosed method need to use values of S which runs also over negative values.
Furthermore the method disclosed in the above mentioned publication uses also a Fourier-Bessel transform. As it is well known Bessel beams constitute a base which can be used to express any function. In the disclosed method integrals running over positive values of S only since the Fourier-Bessel transform cannot accept negative values of the integration variable and integrals running also over negative values of the integration variable are mixed together. This approximation is justified as mentioned in the document only if the shift-term S could tend to infinity. However the paraxial approximation assumed in the above cited document is known as hold only for small values of S. Thus the method disclosed in the above cited publication is limited to configurations of the beam shape and dimensions for which the paraxial approximation is valid. For cases for which the paraxial condition is not met the authors are obliged to carry out numerical simulations. The approximate numerical simulation taught in the above publication aims only to the production of a flat-top beam and the document does not teach how to evaluate amplitudes and phases of the Bessel beams by a closed-form equation and not even by general numerical procedures. Only nan hope of generalisation of the disclosed approach is expressed by the authors.
Thus principally due to the fact that the disclosed method according to the above publication is limited by the role of the negative values of S and by the condition of the paraxial approximation this disclosed method does not allow a general treatment which is based on exact analytical equations which allow a complete control of all the parameters entering into play and of their experimental effects for producing in a simple way and without any approximation a stationary localized wave-field with an a-priori determined arbitrary shape.
A first target of the present invention is to provide a method for generating spatially confined stationary wave fields with any arbitrary (longitudinal) shape in correspondence with a segment of the type 0≦z≦L, quantity L being a periodicity interval which overcomes the drawbacks of the known methods.
A second aim is to provide a method for producing a source which generates the above-said spatially confined stationary wave field.
A further object of the present invention is to provide for an apparatus for generating a spatially confined stationary wave field.